Interleavers for Turbo Codes Using Permutation Polynomials Over Integer Rings Jing Sun, Student Member, IEEE, and Oscar Y

Total Page:16

File Type:pdf, Size:1020Kb

Interleavers for Turbo Codes Using Permutation Polynomials Over Integer Rings Jing Sun, Student Member, IEEE, and Oscar Y IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 1, JANUARY 2005 101 Interleavers for Turbo Codes Using Permutation Polynomials Over Integer Rings Jing Sun, Student Member, IEEE, and Oscar Y. Takeshita, Member, IEEE Abstract—In this paper, a class of deterministic interleavers for code. Further improvements to the S-random interleaver have turbo codes (TCs) based on permutation polynomials over is been presented in [3], [4]. In [3], an iterative decoding suitability introduced. The main characteristic of this class of interleavers (IDS) criterion is used to design two-step S-random interleavers. is that they can be algebraically designed to fit a given compo- nent code. Moreover, since the interleaver can be generated by a In [4], multiple error events are considered. These improved in- few simple computations, storage of the interleaver tables can be terleavers have a better performance than S-random interleavers, avoided. By using the permutation polynomial-based interleavers, especially for short frame sizes. the design of the interleavers reduces to the selection of the coeffi- TCs using random interleavers require an interleaver table to cients of the polynomials. It is observed that the performance of the be stored both in the encoder and the decoder. This is not de- TCs using these permutation polynomial-based interleavers is usu- ally dominated by a subset of input weight P error events. The sirable in some applications, especially when the frame size is minimum distance and its multiplicity (or the first few spectrum large or many different interleavers have to be stored [7]. De- lines) of this subset are used as design criterion to select good per- terministic interleavers can avoid interleaver tables since inter- mutation polynomials. A simple method to enumerate these error leaving and de-interleaving can be performed algorithmically. events for small is presented. The simplest deterministic interleavers are block interleavers Searches for good interleavers are performed. The decoding per- formance of these interleavers is close to S-random interleavers for [8] and linear interleavers [6]. For some component codes long frame sizes. For short frame sizes, the new interleavers out- and very short frame sizes, block and linear interleavers, with perform S-random interleavers. properly chosen parameters, perform better than random inter- Index Terms—Integer ring, interleaver, permutation polyno- leavers. However, for medium to long frame sizes, they have a mial, turbo codes (TCs), weight spectrum. high error floor [6] and random interleavers are still better. In [6], quadratic interleavers have been introduced. Quadratic interleavers are a class of deterministic interleavers based on a I. INTRODUCTION quadratic congruence. For , it can be shown that TYPICAL turbo code (TC) [1] is constructed by parallel A concatenating two convolutional codes via an interleaver. The design of the interleaver is critical to the performance of the TC, particularly for short frame sizes. Interleavers for TCs have been extensively studied in [2]–[6]. They can in general be sep- is a permutation of for odd . Then the arated into two classes: random interleavers and deterministic quadratic interleaver is defined as interleavers. The basic random interleavers permute the information bits in a pseudorandom manner. It was demonstrated in [1] that near- Shannon limit performance can be achieved with these inter- which means that the data in position is interleaved to position leavers for large frame sizes. The S-random interleaver pro- , for all . The quadratic interleavers perform well. posed in [2] is an improvement to the random interleaver. The Even though they are not as good as S-random interleavers, it S-random interleaver is a pseudorandom interleaver with the re- is claimed in [6] that they achieve the average performance of striction that any two input positions within distance cannot random interleavers. Due to this characteristic, in the simula- be permuted to two output positions within distance . Under tions in Section V, in addition to S-random interleavers, we will this restriction, certain short error events in one component code compare our proposed interleavers to quadratic interleavers in- will not be mapped to short error events in the other component stead of the average of many randomly generated interleavers. Recently, some other deterministic interleavers have been proposed. In [9], dithered relative prime (DRP) interleavers are Manuscript received August 20, 2003; revised July 21, 2004. This work was supported by the National Science Foundation under Grants CCR-00752219 used. A good minimum distance property can be achieved for and ANI-9809018. short frame size cases by carefully selecting the parameters J. Sun was with the Electrical and Computer Engineering Department, the of the interleaving process. In [10], monomial interleavers are Ohio State University, Columbus OH 43210 USA. He is now with Qualcomm, San Diego, CA 92121 USA (e-mail: [email protected]). used, which are based on a special permutation polynomial O. Y. Takeshita are with the Electrical and Computer Engineering De- in the form of over a finite field. This class of interleavers partment, the Ohio State University, Columbus OH 43210 USA (e-mail: can also achieve a similar performance as random interleavers. [email protected]). Communicated by Ø. Ytrehus, Associate Editor for Coding Techniques. However, since the permutation polynomials are over a finite Digital Object Identifier 10.1109/TIT.2004.839478 field, the frame size can only be a power of a prime. 0018-9448/$20.00 © 2005 IEEE 102 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 1, JANUARY 2005 In this paper, we will propose another class of deterministic Theorem 2.1. However, since in many of our design examples interleavers. These interleavers are based on permutation poly- we use , we keep Theorem 2.1 for its simplicity. nomials over the ring of integers modulo , . By carefully For general , we also have a necessary and sufficient con- selecting the coefficients of the polynomial, we can achieve dition for a polynomial to be a permutation polynomial, which a performance close to, or in some cases, even better than is summarized in the following theorem. S-random interleavers. Theorem 2.3: For any , where ’s are dis- It has been observed that a subset of error events with input tinct prime numbers, is a permutation polynomial modulo weight , being a small positive integer, usually dominates if and only if is also a permutation polynomial modulo the performance, at least when the frame size is not very short. , . The parameter selection of the permutation polynomial is based Proof: First we prove the necessity. on the minimum distance or the first few spectrum lines of this Let be a permutation polynomial over . For any subset. For small , we present a simple method to find these error events. This helps to reduce the complexity of searching for good permutation polynomial-based interleavers given the component code. The paper is organized as follows. In Section II, we briefly review permutation polynomials over . Interleavers based on permutation polynomials are introduced in Section III. The minimum distance and the spectrum based analysis are given in So . Section IV. Simulation results are shown in Section V, and fi- Assume that is not a permutation polynomial over . nally, we draw our conclusions in Section VI. Then there exist , such that II. PERMUTATION POLYNOMIALS OVER Then Given an integer , a polynomial So a total of numbers are mapped to . This where and are nonnegative integers, is said to be a permutation polynomial over when permutes cannot be true since is a permutation polynomial modulo . In this paper, all the summations and mul- and it can map exactly numbers to . tiplications are modulo unless explicitly stated. We further Next, we prove the sufficiency. Let and be coprime. define the formal derivative of the polynomial to be a poly- Let be a permutation polynomial both modulo and nomial such that . Assume that is not a permutation polynomial modulo . Then there exist and such that (1) and .Define , and , . Since For the special case that , the necessary and sufficient ,wehave condition for a polynomial to be a permutation polynomial is and . Since is a permutation shown in [11]. It is repeated in the following theorem. polynomial modulo and , we must have and . By the Chinese Remainder Theorem Theorem 2.1: Let [13], this implies that , which contradicts be a polynomial with integer coefficients. is a permutation the assumption. Thus, is also a permutation polynomial polynomial over the integer ring if and only if 1) is odd, modulo . 2) is even, and 3) is even. Using this theorem, checking whether a polynomial is a per- For the more general case , where is any prime mutation polynomial modulo reduces to checking the poly- number, the necessary and sufficient condition is derived in [12]. nomial modulo each factor of . Theorem 2.2: is a permutation polynomial over the in- For , we can use Theorem 2.1 to check if the polyno- teger ring if and only if is a permutation polynomial mial is a permutation polynomial, which is a simple test on the over and modulo for all integers . coefficients. For general , we must use Theorem 2.2, which cannot be done by simply looking at the coefficients. In the fol- For example, let and . Then lowing, we will derive some simple check criteria for some sub- , , and , which im- classes of permutation polynomials. plies that is a permutation polynomial over . The If we limit the polynomials to be of second degree, we have formal derivative of is , which the following corollary. is a nonzero constant for all in . Thus, Theorem 2.2 is sat- isfied and is a permutation polynomial over . Corollary 2.4: A second degree polynomial of the form It can be easily verified that when , the necessary is a permutation polynomial over if and and sufficient condition in Theorem 2.2 reduces to the form in only if and .
Recommended publications
  • ON SOME PERMUTATION POLYNOMIALS OVER Fq of THE
    ON SOME PERMUTATION POLYNOMIALS OVER Fq r (q 1)=d OF THE FORM x h(x − ) MICHAEL E. ZIEVE Abstract. Several recent papers have given criteria for certain polynomials to permute Fq, in terms of the periods of certain gen- eralized Lucas sequences. We show that these results follow from a more general criterion which does not involve such sequences. 1. Introduction A polynomial over a finite field is called a permutation polynomial if it permutes the elements of the field. These polynomials first arose in work of Betti [5], Mathieu [25] and Hermite [20] as a way to represent permutations. A general theory was developed by Hermite [20] and Dickson [13], with many subsequent developments by Carlitz and oth- ers. The study of permutation polynomials has intensified in the past few decades, due both to various applications (e.g., [8, 11, 14, 30]) and to an increasing appreciation of the depth of the subtleties inherent to permutation polynomials themselves (for instance, work on permuta- tion polynomials led to a bound on the automorphism group of a curve with ordinary Jacobian [19]). The interesting aspect of permutation polynomials is the interplay between two different ways of representing an object: combinatorially, as a mapping permuting a set, and algebraically, as a polynomial. This is exemplified by one of the first results in the subject, namely that there is no permutation polynomial over Fq of degree q 1 if q > 2 [20]. Much recent work has focused on low-degree permutation− polynomials, as these have quite remarkable properties: for instance, a polynomial of 1=4 degree at most q which permutes Fq will automatically permute Fqn for infinitely many n.
    [Show full text]
  • Constructing Permutation Arrays from Semilinear Groups
    Constructing Permutation Arrays from Semilinear Groups Avi Levy 2/17/2013 Contents 1 Introduction 2 2 Overview and Notation 2 2.1 Finite Fields . 2 2.2 Permutation Polynomials . 2 2.3 Classical Groups . 2 2.4 Hamming distance and Permutation Arrays . 3 3 Constructions 3 3.1 AGL(1; Fq) ............................ 3 3.2 AΓL(1; Fq)............................. 3 3.3 P GL(2; Fq) ............................ 4 3.4 P ΓL(1; Fq)............................. 4 4 Hamming Distance Computations 4 4.1 Preliminaries . 4 4.2 AΓL ................................ 5 4.3 P ΓL ................................ 6 5 Root-counting Results 7 6 Conclusions 10 7 Bibliography 10 1 1 Introduction This is a preliminary writeup of my work with the semilinear groups. These groups yield new permutation arrays with large pairwise Hamming distances. Their properties are well-understood, which allows the minimal pairwise Hamming distance to be obtained without resorting to direct computation. 2 Overview and Notation 2.1 Finite Fields n Given q = p , a power of a prime, call the unique field with q elements Fq. 2 q−2 Fq = f0; 1; g; g ; ··· ; g g where g is any generator of Fq. 2.2 Permutation Polynomials Let f : Fq ! Fq be a polynomial function with coefficients in Fq. If f is one-to-one, then f permutes the elements of Fq. In this case, f is called a permutation polynomial, and the permutation corresponding to f is: σf : x ! f(x) 2.3 Classical Groups There are four families of groups that we are concerned with in this paper. • AGL(n; F): affine general linear group of dimension n over the field F.
    [Show full text]
  • New Permutation Trinomials from Niho Exponents Over Finite Fields
    New Permutation Trinomials From Niho Exponents over Finite Fields with Even Characteristic Nian Li and Tor Helleseth ∗ Abstract In this paper, a class of permutation trinomials of Niho type over finite fields with even char- acteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works. 1 Introduction m Let p be a prime and Fpm denote the finite field with p elements. A polynomial f(x) ∈ Fpm [x] is called a permutation polynomial of Fpm if the associated polynomial function f : c 7→ f(c) from Fpm to Fpm is a permutation of Fpm [15]. Permutation polynomials over finite fields, which were first studied by Hermite [6] and Dickson [2], have wide applications in the areas of mathematics and engineering such as coding theory, cryptography and combinatorial designs. The construction of permutation polynomials with either a simple form or certain desired property is an interesting research problem. For some recent constructions of permutation polynomials with a simple form, the reader is referred to [3, 8, 10, 13, 18, 19, 23] and for some constructions of permutation polynomials with certain desired property, the reader is referred to [1, 4, 5, 12, 20, 21, 22] and the references therein for example. Permutation binomials and trinomials attract researchers’ attention in arXiv:1606.03768v1 [cs.IT] 12 Jun 2016 recent years due to their simple algebraic form, and some nice works on them had been achieved in [3, 10, 18, 23].
    [Show full text]
  • New Permutation Trinomials Constructed from Fractional
    1 New Permutation Trinomials Constructed from Fractional Polynomials Kangquan Li, Longjiang Qu, Chao Li and Shaojing Fu** Abstract Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures. Index Terms Finite Field, Permutation Trinomial, Fractional Polynomial. 1. INTRODUCTION Let Fq be the finite field with q elements. A polynomial f ∈ Fq[x] is called a permutation polynomial (PP) if the induced mapping x → f(x) is a permutation of Fq. The study about permutation polynomials over finite fields attracts people’s interest for many years due to their wide applications in coding theory [15, 24], cryptography [19, 22, 23] and combinatorial designs [5]. For example, Ding et al.[5] constructed a family of skew Hadamard difference sets via the Dickson permutation polynomial of order five, which disproved the longstanding conjecture on skew Hadamard difference sets. More recent progress on permutation polynomials can be found in [2, 4, 9, 10, 12–14, 21, 25, 27]. Permutation trinomials over finite fields are in particular interesting for their simple algebraic form and arXiv:1605.06216v1 [cs.IT] 20 May 2016 additional extraordinary properties.
    [Show full text]
  • Polynomial Functions and Permutation Polynomials Over Some Finite Commutative Rings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Number Theory 105 (2004) 192–202 http://www.elsevier.com/locate/jnt Polynomial functions and permutation polynomials over some finite commutative rings$ Qifan Zhang College of Mathematics, Sichuan University, Chengdu 610064, China Received 18 July 2003 Communicated by S.-W. Zhang Abstract We extend some classical results on polynomial functions mod pl: We prove all results in algebraic methods avoiding any combinatorial calculation. As applications of our methods, we obtain some interesting new results on permutation polynomials in several variables over some finite commutative rings. r 2003 Elsevier Inc. All rights reserved. Keywords: Polynomial function; Permutation polynomial; Witt polynomial; Teichmu¨ ller element 1. Introduction We are interested in polynomial functions and permutation polynomials over finite commutative rings. Without loss of generality, we need only to consider local rings, for example, the ring Z=plZ: The polynomial functions over Z=plZ are widely used in the literature on finite combinatorics (see e.g. [1,2,11]). It is well-known that each function over a finite field is a polynomial function. One should ask when a function f over Z=plZ is a polynomial function. A clear necessary condition, easily l derived from Taylor’s formula, is that there exist functions fi over Z=p Z; i ¼ 0; 1; y; l À 1 such that for any x; sAZ=plZ; the following equality holds: lÀ1 f ðx þ psÞ¼f0ðxÞþf1ðxÞps þ ? þ flÀ1ðxÞðpsÞ : $This work is supported by NSFC (10128103,19901023).
    [Show full text]
  • Permutation Polynomial Interleavers
    1 Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective Oscar Y. Takeshita Dept. of Electrical and Computer Engineering 2015 Neil Avenue The Ohio State University Columbus, OH 43210 [email protected] Submitted to the IEEE Transactions on Information Theory January 12, 2006 Abstract An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic con- structions are important because they admit analytical designs and simple, practical hardware implementation. The spread factor of an interleaver is a common measure for turbo coding applications. Maximum-spread interleavers are interleavers whose spread factors achieve the upper bound. An infinite sequence of quadratic permutation poly- nomials over integer rings that generate maximum-spread interleavers is presented. New properties of permutation polynomial interleavers are investigated from an algebraic-geometric perspective resulting in a new non-linearity metric for interleavers. A new interleaver metric that is a function of both the non-linearity metric and the spread factor is proposed. It is numerically demonstrated that the spread factor has a diminishing importance with the block length. A table of good interleavers for a variety of interleaver lengths according to the new metric is listed. Extensive computer simulation results with impressive frame error rates confirm the efficacy of the new metric. Further, when tail-biting constituent codes are used, the resulting turbo codes are quasi-cyclic. Index Terms algebraic, geometry, interleaver, permutation polynomial, quadratic, quasi-cyclic, spread, turbo code. arXiv:cs/0601048v1 [cs.IT] 12 Jan 2006 2 I. INTRODUCTION Interleavers for turbo codes [1]–[14] have been extensively investigated. However, the design of inter- leavers for turbo codes is complex enough and we believe there are still several relevant open questions.
    [Show full text]
  • A Classification of Permutation Polynomials Through Some Linear
    A Classification of Permutation Polynomials through some linear maps Megha M. Kolhekar, Harish K. Pillai {meghakolhekar, hp}@ee.iitb.ac.in Department of Electrical Engineering, IIT Bombay India November 12, 2019 Abstract In this paper, we propose linear maps over the space of all polyno- mials f(x) in Fq[x] that map 0 to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with per- mutation polynomials. We study certain properties of these linear maps. We propose to classify permutation polynomials by identifying the gen- eralized eigenspaces of these maps, where the permutation polynomials reside. As it turns out, several classes of permutation polynomials stud- ied in literature neatly fall into classes defined using these linear maps. We characterize the shapes of permutation polynomials that appear in the various generalized eigenspaces of these linear maps. For the case of Fp, these generalized eigenspaces provide a degree-wise distribution of poly- nomials (and therefore permutation polynomials) over Fp. We show that for Fq, it is sufficient to consider only a few of these linear maps. The intersection of the generalized eigenspaces of these linear maps contain (permutation) polynomials of certain shapes. In this context, we study a F class of permutation polynomials over p2 . We show that the permutation polynomials in this class are closed under compositional inverses. We also do some enumeration of permutation polynomials of certain shapes. arXiv:1911.03689v1 [math.NT] 9 Nov 2019 1 Introduction and Related Work Permutation polynomials (PPs) are important in cryptography, combina- torics and coding theory, to name a few areas.
    [Show full text]
  • Constructions of Rank Modulation Codes That Are Useful for Error Correction in Powerline Communication
    1 Constructions of Rank Modulation Codes Arya Mazumdar∗, Alexander Barg§ and Gilles Z´emora Abstract—Rank modulation is a way of encoding information the drift in different cells may occur at different speed, errors to correct errors in flash memory devices as well as impulse introduced in the data are adequately accounted for by tracking noise in transmission lines. Modeling rank modulation involves the relative value of adjacent cells rather than the absolute construction of packings of the space of permutations equipped with the Kendall tau distance. values of cell charges. Storing information in relative values We present several general constructions of codes in permuta- of the charges also simplifies the rewriting of the data because tions that cover a broad range of code parameters. In particular, we do not need to reach any particular value of the charge as we show a number of ways in which conventional error-correcting long as we have the desired ranking, thereby reducing the risk codes can be modified to correct errors in the Kendall space. of overprogramming. Based on these ideas, Jiang et al. [14], Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to [15] suggested to use the rank modulation scheme for error- correct errors in the Hamming metric. For instance, from binary correcting coding of data in flash memories. A similar noise BCH codes we obtain codes correcting t Kendall errors in n model arises in transmission over channels subject to impulse t memory cells that support the order of n!/(log2 n!) messages, noise that changes the value of the signal substantially but for any constant t = 1, 2,...
    [Show full text]
  • PERMUTATION BINOMIALS OVER FINITE FIELDS 1. Introduction A
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 8, August 2009, Pages 4169–4180 S 0002-9947(09)04578-4 Article electronically published on March 17, 2009 PERMUTATION BINOMIALS OVER FINITE FIELDS ARIANE M. MASUDA AND MICHAEL E. ZIEVE Abstract. m n F We prove that if x + ax permutes the√ prime field p,where ∈ F∗ − − − m>n>0anda p,thengcd(m n, p 1) > p 1. Conversely, we prove that if q ≥ 4andm>n>0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over Fq of the form xm + axn if and only if gcd(m, n, q − 1) = 1. 1. Introduction A polynomial over a finite field is called a permutation polynomial if it permutes the elements of the field. These polynomials first arose in work of Betti [1] and Hermite [10] as a way to represent permutations. A general theory was developed by Hermite [10] and Dickson [6], with many subsequent developments by Carlitz and others. The simplest class of nonconstant polynomials are the monomials m m x with m>0, and one easily checks that x permutes Fq if and only if m is coprime to q − 1. However, already for binomials the situation becomes much more mysterious. Some examples occurred in Hermite’s work [10], and Mathieu [17] pi i showed that x − ax permutes Fq whenever a is not a (p − 1)-th power in Fq;here p denotes the characteristic of Fq. A general nonexistence result was proved by Niederreiter and Robinson [20] and improved by Turnwald [28]: m n F ∈ F∗ Theorem 1.1.
    [Show full text]
  • The Structure of a Group of Permutation Polynomials
    J. Austral. Math. Soc. (Series A) 38 (1985), 164-170 THE STRUCTURE OF A GROUP OF PERMUTATION POLYNOMIALS GARY L. MULLEN and HARALD MEDERREITER (Received 24 November 1983; revised 31 January 1984) Communicated by R. Lidl Abstract Let Gq be the group of permutations of the finite field Fq of odd order q that can be represented by iq+v>/2 polynomials of the form ax + bx with a, b e Fq. It is shown that Gq is isomorphic to the regular wreath product of two cyclic groups. The structure of Gq can also be described in terms of cyclic, dicyclic, and dihedral groups. It also turns out that Gq is isomorphic to the symmetry group of a regular complex polygon. 1980 Mathematics subject classification (Amer. Math. Soc): 12 C 05, 20 B 25, 20 E 22, 51 F 15 1. Introduction Let F be the finite field of order q. Then every mapping from Fq into itself can be uniquely represented by a polynomial in Fq[x] of degree less than q, and composition of mappings corresponds to composition of polynomials mod(jc* — x) (see [9, Chapter 7]). In particular, every group of permutations of Fq can be represented by a set of polynomials in Fq[x] of degree less than q that is closed under composition vao&(xq — x). According to a well-known definition (see [8, Chapter 4], [9, Chapter 7]), a polynomial / over Fq for which the corresponding polynomial mapping c e Fq -* f(c) is a permutation is called a permutation polynomial of Fq. Numerous papers have been written on the structure of permutation groups represented by a given group of permutation polynomials of F under composition mod(x'7 — x); see for example Carlitz [1], > 1985 Australian Mathematical Society 0263-6115/85 $A2.00 + 0.00 164 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • 8.3 Value Sets of Polynomials
    Permutation polynomials 225 arbitrary subfields of the finite field. The results developed in this paper are used to con- struct additional complete sets of frequency squares, rectangles and hyperrectangles, and orthogonal arrays. Some of the theorems present a relationship between permutation poly- nomials of a finite field and field permutation functions, an implicit bound on the possible number of functions in an orthogonal field system, and results related to the generation of orthogonal field systems. 8.2.21 Remark For finite rings there are two concepts to distinguish: permutation polynomials (as above) and strong permutation polynomials. The latter are defined via the cardinality of the inverse image. Polynomials are strong permutation polynomials (or strong orthogonal systems) in n variables if they can be completed to an orthogonal system of n polynomials in n variables. 8.2.22 Theorem [1129] If R is a local ring whose maximal ideal has a minimal number m of generators, then for every n > m there exists a permutation polynomial in n variables that is not a strong permutation polynomial. 8.2.23 Corollary [1129] Every permutation polynomial in any number of variables over a local ring R is strong if and only if R is a finite field. 8.2.24 Remark Wei and Zhang showed [2954] that if n m in the setting of Theorem 8.2.22, then every orthogonal system of k polynomials in n variables≤ can be completed to an orthogonal system of n polynomials (and in particular, every permutation polynomial is strong). See Also 9.4 Considers κ-polynomials used for constructions of semifields.
    [Show full text]
  • Quadratic Permutation Polynomial Interleavers for LTE Turbo Coding
    International Journal of Future Computer and Communication, Vol. 2, No. 4, August 2013 Quadratic Permutation Polynomial Interleavers for LTE Turbo Coding Ching-Lung Chi Abstract—Interleaver is an important component of turbo min δ, (2) , code and has a strong impact on its error correcting performance. Quadratic permutation polynomial (QPP) δ, is the Lee metric between points interleaver is a contention-free interleaver which is suitable for , π and p j,πj parallel turbo decoder implementation. To improve the performance of LTE QPP interleaver, largest-spread and , || | | (3) maximum-spread QPP interleavers are analyzed for Long Term Evolution (LTE) turbo codes in this paper. Compared where with LTE QPP interleaver, those algorithms have low computing complexity and better performances from || mod , mod (4) simulation results. The quadratic polynomials which lead to the largest spreading factor for some interleaver lengths are given in Index Terms—Quadratic permutation polynomial (QPP) interleaver, Long Term Evolution (LTE), largest-spread QPP [2]. An algorithm for faster computation of is also interleaver, maximum-spread QPP interleaver. presented. It is based on the representatives of orbits in the representation of interleavercode. Section II gives a brief review of QPP interleaver. Largest-Spread and Maximum-Spread QPP interleavers are I. INTRODUCTION the topics of Section III. Section IV presents the bit error The selection of turbo coding was considered during the rates (BER) resulted from simulations for LTE standard, LS study phase of Long Term Evolution (LTE) standard to and MS QPP inter leavers length 512. Section V concludes meet the stringent requirements. Following deliberations the paper. within the working group, the quadratic permutation polynomials (QPP) were selected as interleavers for turbo codes, emerging as the most promising solutions to the LTE II.
    [Show full text]